The bosonic Hubbard model on a three dimensional flat band lattice

Andreas Mielke; Leon Haag-Fank

arxiv: 2604.19703 · v1 · submitted 2026-04-21 · 🧮 math-ph · cond-mat.stat-mech· math.CO· math.MP

The bosonic Hubbard model on a three dimensional flat band lattice

Leon Haag-Fank , Andreas Mielke This is my paper

Pith reviewed 2026-05-10 01:19 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechmath.COmath.MP

keywords bosonic Hubbard modelflat bandline graphcubic latticeground state entropysubextensive entropy4-cycle decompositions

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The pith

Exact ground states for repulsive bosons on a 3D flat-band lattice exist up to a critical particle number where entropy turns subextensive.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how the line graph of the periodic cubic lattice supports a highly degenerate flat band of localized single-particle states. Repulsive bosons can be placed into these states without overlap to form exact many-body ground states up to a critical filling Nc. At Nc the number of such configurations yields a ground-state entropy that grows only as Nc to the power 2/3, while at lower densities the entropy grows extensively with system size. The counting problem maps onto enumerating 4-cycle decompositions of the cubic lattice. A reader would care because the result gives an exact, solvable example of how flat bands produce unusual degeneracy scaling that affects the thermodynamics of interacting bosons.

Core claim

We construct exact multi-particle ground states of the repulsive bosonic Hubbard model on the line graph of the cubic lattice with periodic boundary conditions by occupying disjoint localized flat-band states up to a critical particle number Nc. At this filling the ground-state entropy is subextensive and proportional to Nc^{2/3}. For lower particle numbers the entropy is extensive. The degeneracy counting is equivalent to the number of 4-cycle decompositions of the cubic lattice.

What carries the argument

Localized flat-band eigenstates of the hopping matrix on the line graph of the cubic lattice, used to build non-overlapping multi-particle configurations whose degeneracy counts 4-cycle decompositions.

If this is right

Exact ground states exist for all fillings up to Nc by non-overlapping occupation of localized flat-band states.
Ground-state entropy is extensive below Nc and subextensive at Nc.
The number of ground states equals the number of 4-cycle decompositions of the cubic lattice with periodic boundaries.
Thermodynamic quantities derived from the ground-state manifold inherit the subextensive scaling at the critical filling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The subextensive entropy may produce a vanishing contribution to the specific heat per particle as system size grows at the critical density.
Similar constructions on other line graphs or higher-dimensional flat-band lattices could yield comparable entropy scalings tied to cycle decompositions.
The mapping to 4-cycle counting suggests that combinatorial algorithms for cycle decompositions could be used to compute the exact degeneracy for finite systems.

Load-bearing premise

The multi-particle states formed by placing bosons into disjoint localized flat-band states are the true lowest-energy eigenstates of the full interacting Hamiltonian.

What would settle it

Exact diagonalization on small periodic systems that checks whether the constructed states remain the ground states and whether their degeneracy scales as predicted with Nc.

Figures

Figures reproduced from arXiv: 2604.19703 by Andreas Mielke, Leon Haag-Fank.

**Figure 1.** Figure 1: Visualization of a tower decomposition in a 3x3x3 sub-lattice of a cubic lattice. The three sets of towers [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

**Figure 2.** Figure 2: The two rotations of a column of a tower decomposition. The faces are labelled [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Left: The center vertex must have exactly one face per direction attached to it. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

read the original abstract

The lowest eigenstates of the hopping matrix on the line graph of a cubic lattice with periodic boundary conditions are highly degenerate, they form a lowest flat band. Further, these states are localized. If one considers a repulsive bosonic Hubbard model on this lattice it is possible to construct exact multi-particle ground states simply by putting particles in the localized single particle ground states such that they avoid each other. This can be done up to a certain critical particle number $N_c$. We prove that at this particle number the ground state entropy is subextensive $\propto N_c^{2/3}$. For lower densities the entropy is extensive. We further show that the problem is related to the number of 4-cycle decompositions of the cubic lattice with periodic boundary conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Exact ground states for the bosonic Hubbard model on a 3D flat-band lattice, with subextensive degeneracy at critical filling via 4-cycle counting.

read the letter

The main takeaway is that this paper constructs exact ground states for repulsive bosons on the line graph of the periodic cubic lattice and shows the ground-state entropy at the highest packing density is subextensive, scaling as N_c^{2/3} rather than extensive in volume. They do this by occupying disjoint localized single-particle flat-band states up to a critical number N_c, after which overlaps are forced. The degeneracy at N_c equals the number of 4-cycle decompositions of the cubic lattice, which they bound combinatorially. For lower densities the entropy is extensive, as usual. The construction works for any U > 0 because the flat band sits at the bottom of the spectrum and the interaction is nonnegative, so these states achieve the global energy minimum with no lower states possible from mixing or higher bands. This is new. Three-dimensional flat-band examples with explicit interacting ground states and a concrete entropy bound are rare; most prior work stays in two dimensions or stays single-particle. The link to 4-cycle decompositions gives a clean way to get the subextensive scaling. The argument is direct and does not rely on fitting or uncontrolled approximations. The soft spots are limited. The combinatorial mapping and bound depend on the periodic cubic geometry and the specific localized states of this line graph, so they do not immediately extend to other lattices or open boundaries. The paper gives no small-system numerics or explicit counts for small N_c to cross-check the asymptotics, though the analytic lower-bound argument on energy is self-contained. A reader working on flat-band systems, exact solutions in Hubbard models, or combinatorial aspects of lattice ground states will find the explicit states and the entropy result useful as a benchmark. It is the sort of paper that supplies a concrete starting point rather than a broad claim. The central claims rest on verifiable single-particle localization plus positivity of the interaction, with no load-bearing gaps. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the bosonic Hubbard model on the line graph of the three-dimensional cubic lattice with periodic boundary conditions. It shows that the single-particle hopping matrix possesses a highly degenerate flat band at its lower edge consisting of localized states. For repulsive on-site interactions, exact many-body ground states are constructed by occupying mutually disjoint localized single-particle states up to a critical particle number Nc; at this filling the ground-state degeneracy (entropy) is proven to be subextensive and scales as Nc^{2/3}, while remaining extensive at lower densities. The counting problem is shown to be equivalent to enumerating 4-cycle decompositions of the periodic cubic lattice.

Significance. If the proofs are complete, the work supplies a rare exact result on the ground-state manifold and entropy of an interacting bosonic model in three dimensions. The construction is parameter-free for any U>0, the states achieve the global energy lower bound by independent kinetic and interaction estimates, and the subextensive scaling connects many-body physics to a concrete combinatorial problem on the cubic lattice. Such rigorous degeneracy bounds are valuable benchmarks for flat-band and frustrated systems.

major comments (2)

[many-body construction (around the statement following Eq. for the full Hamiltonian)] The central claim that the constructed states are exact ground states for arbitrary U>0 rests on the interaction term being non-negative and vanishing precisely when supports are disjoint. The manuscript should explicitly verify in the many-body section that no linear combination involving higher-band components or overlapping clusters can reach the same energy; a short argument using the spectral gap of the hopping matrix or the positive-semidefiniteness of the interaction operator would strengthen this.
[combinatorial section on 4-cycle decompositions] The subextensive entropy bound ∝ Nc^{2/3} is load-bearing and relies on both upper and lower combinatorial estimates for the number of maximal 4-cycle decompositions. The manuscript equates the degeneracy to this counting problem but should state the precise lemma or theorem that establishes the O(Nc^{2/3}) upper bound on the logarithm; without an explicit reference to the lattice tiling or matching argument used, the scaling cannot be independently verified.

minor comments (3)

[Introduction] The definition of the line graph and the precise embedding of the cubic lattice with periodic boundary conditions should be recalled with a short figure or coordinate description in the introduction for readers unfamiliar with graph-theoretic formulations of lattice models.
[single-particle spectrum] Notation for the single-particle localized states (e.g., their support on plaquettes or edges) is introduced but could be made uniform across sections; a table summarizing the support size and orthogonality relations would improve readability.
[Introduction] A brief comparison with known flat-band Hubbard models in lower dimensions (e.g., the kagome or Lieb lattice) would help situate the three-dimensional result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We address the major comments point by point below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: [many-body construction (around the statement following Eq. for the full Hamiltonian)] The central claim that the constructed states are exact ground states for arbitrary U>0 rests on the interaction term being non-negative and vanishing precisely when supports are disjoint. The manuscript should explicitly verify in the many-body section that no linear combination involving higher-band components or overlapping clusters can reach the same energy; a short argument using the spectral gap of the hopping matrix or the positive-semidefiniteness of the interaction operator would strengthen this.

Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will add a short paragraph in the many-body section. We will use the spectral gap of the hopping matrix above the flat band together with the positive-semidefiniteness of the interaction operator (which vanishes if and only if the supports are pairwise disjoint) to show that any linear combination containing higher-band components or overlapping localized states necessarily has strictly higher energy. This confirms that the constructed states achieve the global lower bound for any U>0. revision: yes
Referee: [combinatorial section on 4-cycle decompositions] The subextensive entropy bound ∝ Nc^{2/3} is load-bearing and relies on both upper and lower combinatorial estimates for the number of maximal 4-cycle decompositions. The manuscript equates the degeneracy to this counting problem but should state the precise lemma or theorem that establishes the O(Nc^{2/3}) upper bound on the logarithm; without an explicit reference to the lattice tiling or matching argument used, the scaling cannot be independently verified.

Authors: We thank the referee for this request for added clarity. In the revised version we will explicitly state the precise lemma (or theorem) that supplies the O(Nc^{2/3}) upper bound on the logarithm of the number of maximal 4-cycle decompositions. We will also include a direct reference to the lattice tiling or matching argument employed in our combinatorial analysis, so that the subextensive scaling can be verified independently. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper directly constructs the flat-band localized single-particle eigenstates of the hopping matrix on the line graph of the periodic cubic lattice, establishes that the interaction term is nonnegative, and shows that any collection of mutually disjoint-support localized states therefore saturates the global energy lower bound (N times flat-band energy plus zero interaction) for arbitrary U>0. The ground-state degeneracy at filling N_c is then identified with the number of maximal such collections, which the paper equates to the number of 4-cycle decompositions of the lattice and bounds combinatorially as exp(O(N_c^{2/3})). No fitted parameters are renamed as predictions, no load-bearing step reduces to a self-citation chain, and the combinatorial equivalence is an independent counting argument rather than a definitional loop. The central claims therefore rest on explicit lower bounds and lattice combinatorics that do not presuppose the target entropy result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard properties of line graphs of cubic lattices and the existence of localized flat-band eigenstates; no free parameters or new entities are introduced.

axioms (2)

domain assumption The lowest eigenstates of the hopping matrix on the line graph of the cubic lattice with periodic boundary conditions form a highly degenerate flat band and can be chosen localized.
Invoked as the starting point for the exact multi-particle construction.
domain assumption Repulsive on-site interactions make any state with overlapping particles higher in energy than non-overlapping placements.
Used to establish that the constructed states are ground states.

pith-pipeline@v0.9.0 · 5429 in / 1429 out tokens · 35963 ms · 2026-05-10T01:19:07.255935+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 8 canonical work pages

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Andreas Mielke and Hal Tasaki. “Ferromagnetism in the Hubbard model - Examples from Models with Degenerate Single-Electron Ground States”. In:Commun. Math. Phys.158.2 (1993). _eprint: 9305026v2, pp. 341–371.issn: 0010-3616.doi:10.1007/BF02108079.url:http://www.springerlink.com/index/10. 1007/BF02108079

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Universal d= 1 flat band generator from compact localized states

Wulayimu Maimaiti, Sergej Flach and Alexei Andreanov. “Universal d= 1 flat band generator from compact localized states”. In:Physical Review B99.12 (2019). Publisher: APS, p. 125129

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Flat-bandgeneratorintwodimensions

WulayimuMaimaiti,AlexeiAndreanovandSergejFlach.“Flat-bandgeneratorintwodimensions”.In:Physical Review B103.16 (2021). Publisher: APS, p. 165116

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Non-Hermitian flat-band generator in one dimension

Wulayimu Maimaiti and Alexei Andreanov. “Non-Hermitian flat-band generator in one dimension”. In:Phys- ical Review B104.3 (2021). Publisher: APS, p. 035115

2021
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An Extension of the Cell-Construction Method for the Flat-Band Ferromagnetism

Akinori Tanaka. “An Extension of the Cell-Construction Method for the Flat-Band Ferromagnetism”. In:J. Stat. Phys.181 (2020), pp. 897–916.doi:10.1007/s10955-020-02610-3.url:https://arxiv.org/pdf/ 2004.07516. 6

work page doi:10.1007/s10955-020-02610-3.url:https://arxiv.org/pdf/ 2020
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Bose-Hubbard model on two-dimensional line graphs

Johannes Motruk and Andreas Mielke. “Bose-Hubbard model on two-dimensional line graphs”. In:J. Phys. A45.22 (Apr. 2012). _eprint: 1112.0131v3, p. 225206.doi:10 . 1088 / 1751 - 8113 / 45 / 22 / 225206.url: http://arxiv.org/pdf/1112.0131v3

work page arXiv 2012
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Pair formation of hard core bosons in flat band systems

Andreas Mielke. “Pair formation of hard core bosons in flat band systems”. In:J Stat Phys.171 (2018), pp. 679–695.url:https://doi.org/10.1007/s10955-018-2030-0

work page doi:10.1007/s10955-018-2030-0 2018
[16]

Localised pair formation in bosonic flat-band Hubbard models

Jacob Fronk and Andreas Mielke. “Localised pair formation in bosonic flat-band Hubbard models”. In:J. Stat. Phys.185 (2021), p. 14

2021
[17]

Decomposition of product graphs into paths and cycles of length four

S. Jeevadoss and A. Muthusamy. “Decomposition of product graphs into paths and cycles of length four”. In: Graphs and Combinatorics32 (2015), pp. 199–233

2015
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Cycle decompositions of the Cartesian product of cycles

S. A. Tapadia, Y. M. Borse and B.N.Waphare. “Cycle decompositions of the Cartesian product of cycles”. In: Australas. J. Combin.74 (2019), pp. 443–459.url:https://ajc.maths.uq.edu.au/pdf/74/ajc_v74_ p443.pdf

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Huijse and K

L. Huijse and K. Schoutens.Quantum phases of supersymmetric lattice models. 2009.url:https://arxiv. org/pdf/0910.2386

work page arXiv 2009
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Supersymmetry, lattice fermions, independence complexes and cohomo- logy theory

Liza Huijse and Kareljan Schoutens. “Supersymmetry, lattice fermions, independence complexes and cohomo- logy theory”. In:Adv. Theor. Math. Phys.14 (2010), pp. 643–694

2010
[21]

Dense Packing on Uniform Lattices

Kari Eloranta. “Dense Packing on Uniform Lattices”. In:Journal of Statistical Physics130: (2008), pp. 741– 755.doi:10.1007/s10955-007-9448-0

work page doi:10.1007/s10955-007-9448-0 2008
[22]

Phase diagram and critical behavior of the square-lattice Ising model with com- peting nearest-neighbor and next-nearest-neighbor interactions

Junqi Yin and D.P. Landau. “Phase diagram and critical behavior of the square-lattice Ising model with com- peting nearest-neighbor and next-nearest-neighbor interactions”. In:Physical Review E80 (2009), p. 051117. doi:10.1103/PhysRevE.80.051117. 7 One tower for each color All additional towers, that fit in the sub-lattice All towers required to occupy al...

work page doi:10.1103/physreve.80.051117 2009

[1] [1]

Electron correlations in narrow energy bands

J Hubbard. “Electron correlations in narrow energy bands”. In:Proc. Roy. Soz. A276 (1963), p. 238

1963

[2] [2]

Electron Correlation and Ferromagnetism of Transition Metals

J Kanamori. “Electron Correlation and Ferromagnetism of Transition Metals”. In:Prog. Theo. Phys.30 (1963), p. 275

1963

[3] [3]

Effect of correlation on the ferromagnetism of transition metals

Martin C Gutzwiller. “Effect of correlation on the ferromagnetism of transition metals”. In:Phys. Rev. Lett.10.5 (1963). Publisher: APS, p. 159.url:http : / / journals . aps . org / prl / abstract / 10 . 1103 / PhysRevLett.10.159

1963

[4] [4]

Quantum Cell Model for Boson

H. A. Gersch and G. C. Knollman. “Quantum Cell Model for Boson”. In:Phys. Rev.129 (1963), pp. 959–967

1963

[5] [5]

The Hubbard Model and its Properties

Andreas Mielke. “The Hubbard Model and its Properties”. In:Modeling and Simulation5 (2015). Backup Publisher: Verlag des Forschungszentrum Jülich ISBN: 978-3-95806-074-6 Publisher: Eva Pavarini, Erik Koch, and Piers Coleman, pp. 1–26.url:http://www.cond-mat.de/events/correl15/manuscripts/mielke.pdf

2015

[6] [6]

Texier and G

A Mielke. “Ferromagnetism in the Hubbard model on line graphs and further considerations”. In:J. Phys. A: Math. Gen.24.14 (1991), pp. 3311–3321.issn: 0305-4470.doi:10.1088/0305- 4470/24/14/018.url: http://stacks.iop.org/0305-4470/24/3311

work page doi:10.1088/0305- 1991

[7] [7]

Ferromagnetism in the Hubbard Models with Degenerate Single-Electron Ground States

H Tasaki. “Ferromagnetism in the Hubbard Models with Degenerate Single-Electron Ground States”. In:Phys. Rev. Lett.69 (1992), p. 1608

1992

[8] [8]

Ferromagnetism in the Hubbard model - Examples from Models with Degenerate Single-Electron Ground States

Andreas Mielke and Hal Tasaki. “Ferromagnetism in the Hubbard model - Examples from Models with Degenerate Single-Electron Ground States”. In:Commun. Math. Phys.158.2 (1993). _eprint: 9305026v2, pp. 341–371.issn: 0010-3616.doi:10.1007/BF02108079.url:http://www.springerlink.com/index/10. 1007/BF02108079

work page doi:10.1007/bf02108079.url:http://www.springerlink.com/index/10 1993

[9] [9]

Compact localized states and flat-band generators in one dimension

Wulayimu Maimaiti et al. “Compact localized states and flat-band generators in one dimension”. In:Physical Review B95.11 (2017). Publisher: APS, p. 115135

2017

[10] [10]

Universal d= 1 flat band generator from compact localized states

Wulayimu Maimaiti, Sergej Flach and Alexei Andreanov. “Universal d= 1 flat band generator from compact localized states”. In:Physical Review B99.12 (2019). Publisher: APS, p. 125129

2019

[11] [11]

Flat-bandgeneratorintwodimensions

WulayimuMaimaiti,AlexeiAndreanovandSergejFlach.“Flat-bandgeneratorintwodimensions”.In:Physical Review B103.16 (2021). Publisher: APS, p. 165116

2021

[12] [12]

Non-Hermitian flat-band generator in one dimension

Wulayimu Maimaiti and Alexei Andreanov. “Non-Hermitian flat-band generator in one dimension”. In:Phys- ical Review B104.3 (2021). Publisher: APS, p. 035115

2021

[13] [13]

An Extension of the Cell-Construction Method for the Flat-Band Ferromagnetism

Akinori Tanaka. “An Extension of the Cell-Construction Method for the Flat-Band Ferromagnetism”. In:J. Stat. Phys.181 (2020), pp. 897–916.doi:10.1007/s10955-020-02610-3.url:https://arxiv.org/pdf/ 2004.07516. 6

work page doi:10.1007/s10955-020-02610-3.url:https://arxiv.org/pdf/ 2020

[14] [14]

Bose-Hubbard model on two-dimensional line graphs

Johannes Motruk and Andreas Mielke. “Bose-Hubbard model on two-dimensional line graphs”. In:J. Phys. A45.22 (Apr. 2012). _eprint: 1112.0131v3, p. 225206.doi:10 . 1088 / 1751 - 8113 / 45 / 22 / 225206.url: http://arxiv.org/pdf/1112.0131v3

work page arXiv 2012

[15] [15]

Pair formation of hard core bosons in flat band systems

Andreas Mielke. “Pair formation of hard core bosons in flat band systems”. In:J Stat Phys.171 (2018), pp. 679–695.url:https://doi.org/10.1007/s10955-018-2030-0

work page doi:10.1007/s10955-018-2030-0 2018

[16] [16]

Localised pair formation in bosonic flat-band Hubbard models

Jacob Fronk and Andreas Mielke. “Localised pair formation in bosonic flat-band Hubbard models”. In:J. Stat. Phys.185 (2021), p. 14

2021

[17] [17]

Decomposition of product graphs into paths and cycles of length four

S. Jeevadoss and A. Muthusamy. “Decomposition of product graphs into paths and cycles of length four”. In: Graphs and Combinatorics32 (2015), pp. 199–233

2015

[18] [18]

Cycle decompositions of the Cartesian product of cycles

S. A. Tapadia, Y. M. Borse and B.N.Waphare. “Cycle decompositions of the Cartesian product of cycles”. In: Australas. J. Combin.74 (2019), pp. 443–459.url:https://ajc.maths.uq.edu.au/pdf/74/ajc_v74_ p443.pdf

2019

[19] [19]

Huijse and K

L. Huijse and K. Schoutens.Quantum phases of supersymmetric lattice models. 2009.url:https://arxiv. org/pdf/0910.2386

work page arXiv 2009

[20] [20]

Supersymmetry, lattice fermions, independence complexes and cohomo- logy theory

Liza Huijse and Kareljan Schoutens. “Supersymmetry, lattice fermions, independence complexes and cohomo- logy theory”. In:Adv. Theor. Math. Phys.14 (2010), pp. 643–694

2010

[21] [21]

Dense Packing on Uniform Lattices

Kari Eloranta. “Dense Packing on Uniform Lattices”. In:Journal of Statistical Physics130: (2008), pp. 741– 755.doi:10.1007/s10955-007-9448-0

work page doi:10.1007/s10955-007-9448-0 2008

[22] [22]

Phase diagram and critical behavior of the square-lattice Ising model with com- peting nearest-neighbor and next-nearest-neighbor interactions

Junqi Yin and D.P. Landau. “Phase diagram and critical behavior of the square-lattice Ising model with com- peting nearest-neighbor and next-nearest-neighbor interactions”. In:Physical Review E80 (2009), p. 051117. doi:10.1103/PhysRevE.80.051117. 7 One tower for each color All additional towers, that fit in the sub-lattice All towers required to occupy al...

work page doi:10.1103/physreve.80.051117 2009